This application claims the priorities of Japanese Patent Application No. 2001-23200 filed on Jan. 31, 2001 and Japanese Patent Application No. 2001-399179 filed on Dec. 28, 2001, which are incorporated herein by reference.
1. Field of the Invention
The present invention relates to a phase shift fringe analysis method using Fourier transform when analyzing a fringe image by using a phase shift method, and an apparatus using the same. In particular, the present invention relates to a phase shift fringe analysis method comprising the steps of obtaining image information of interference fringes and the like while shifting a phase by using a phase shift device such as PZT (piezoelectric device), and analyzing thus obtained plurality of image data items having a fringe pattern of interference fringes and the like, thereby attaining highly accurate phase information of an object to be observed; and an apparatus using the same.
2. Description of the Prior Art
While light-wave interference method, for example, has conventionally been known as important means concerning precise measurement of object surfaces, there have recently been urgent needs for developing an interferometry technique (sub-fringe interferometry) for reading out information from a fraction of a single interference fringe (one fringe) or less from the necessity to measure a surface or wavefront aberration of {fraction (1/10)} wavelength or higher.
An example of typical techniques widely in practice as such sub-fringe interferometry is a phase shift fringe analysis method (also known as fringe scanning method or phase scanning method) disclosed in xe2x80x9cPHASE-MEASUREMENT INTERFEROMETRY TECHNIQUES,xe2x80x9d PROGRESS IN OPTICS, VOL. XXVI (1988), pp. 349-393.
In the phase shift method, a phase shift device such as a PZT (piezoelectric) device is used for phase-shifting the relative relationship between an object to be observed and a reference, interference fringe data is captured each time a predetermined step amount shifts, so as to measure the interference fringe intensity of each point of the object surface, and the phase of each point of the surface is determined by using the results of measurement.
In the case of a 4-step phase shift method, for example, respective interference fringe intensities I1, I2, I3, and I4 at individual phase shift steps are expressed as follows:
I1(x, y)=I0(x, y){1+xcex3(x, y)cos[xcfx86(x, y)]}
I2(x, y)=I0(x, y){1+xcex3(x, y)cos[xcfx86(x, y)+xcfx80/2]}xe2x80x83xe2x80x83(2) 
I3(x, y)=I0(x, y){1+xcex3(x, y)cos[xcfx86(x, y)+xcfx80]}
I4(x, y)=I0(x, y){1+xcex3(x, y)cos[xcfx86(x, y)+3xcfx80/2]}
where
x and y are coordinates;
xcfx86 (x, y) is a phase;
I0(x, y) is the average optical intensity at each point; and
xcex3 (x, y) is the modulation of interference fringes.
From these expressions, the phase xcfx86 (x, y) can be determined as                               φ          ⁡                      (                          x              ,              y                        )                          =                              tan                          -              1                                ⁢                                                                                          I                    4                                    ⁡                                      (                                          x                      ,                      y                                        )                                                  -                                                      I                    2                                    ⁡                                      (                                          x                      ,                      y                                        )                                                                                                                    I                    1                                    ⁡                                      (                                          x                      ,                      y                                        )                                                  -                                                      I                    3                                    ⁡                                      (                                          x                      ,                      y                                        )                                                                        .                                              (        3        )            
Though the phase shift method enables measurement with a very high accuracy if a predetermined step amount can shift accurately, it has been problematic in terms of measurement errors occurring due to errors in step amount and in that it is likely to be affected by disturbances during measurement since it necessitates a plurality of interference fringe image data items.
As a sub-fringe interferometry technique other than the phase shift method, attention has been focused on one using a Fourier transform method, for example, as disclosed in xe2x80x9cBasics of Sub-fringe Interferometry,xe2x80x9d Kogaku, Vol. 13, No. 1 (February 1984), pp. 55 to 65.
The Fourier transform fringe analysis method is a technique in which a carrier frequency (caused by a relative tilt between an object surface to be observed and a reference surface) is introduced, whereby the phase of the object can be determined with a high accuracy from a single fringe image. When a carrier frequency is introduced without the initial phase of the object being taken into consideration, the interference fringe intensity i(x, y) is represented by the following expression (4):
i(x, y)=a(x, y)+b(x, y)cos[2xcfx80ƒxx+2xcfx80ƒyy+xcfx86(x, y)]xe2x80x83xe2x80x83(4) 
where
a(x, y) is the background of interference fringes;
b(x, y) is the visibility of fringes;
xcfx86 (x, y) is the phase of the object; and
fx and fy are respective carrier frequencies in x and y directions expressed by:             f      x        =                            2          ·          tan                ⁢                  xe2x80x83                ⁢                  θ          x                    λ        ,      xe2x80x83    ⁢            f      y        =                            2          ·          tan                ⁢                  xe2x80x83                ⁢                  θ          y                    λ      
where xcex is the wavelength of light, and xcex8x and xcex8y are respective inclinations of the object surface in x and y directions.
The above-mentioned expression (4) can be deformed as the following expression (5):
i(x, y)=a(x, y)+c(x, y)exp[i(2xcfx80ƒx+2xcfx80ƒy)]+c*(x, y)exp[xe2x88x92i(2xcfx80ƒx+2xcfx80ƒy)]xe2x80x83xe2x80x83(5) 
where c(x, y) is the complex amplitude of interference fringes, and c*(x, y) is the complex conjugate of c(x, y).
Here, c(x, y) is represented as the following expression (6):                               c          ⁡                      (                          x              ,              y                        )                          =                                                            b                ⁡                                  (                                      x                    ,                    y                                    )                                            ⁢                              exp                ⁡                                  [                                      ⅈ                    ⁢                                          xe2x80x83                                        ⁢                                          φ                      ⁡                                              (                                                  x                          ,                          y                                                )                                                                              ]                                                      2                    .                                    (        6        )            
When the above-mentioned expression (5) is Fourier-transformed, the following expression (7) is obtained:
I(xcex7,xcex6)=A(xcex7,xcex6)+C(xcex7xe2x88x92ƒx,xcex6xe2x88x92ƒy)+C*(xcex7+ƒx,xcex6+ƒy)xe2x80x83xe2x80x83(7) 
where
A(xcex7, xcex6) is the Fourier transform of a(x, y);
C(xcex7xe2x88x92fx, xcex6xe2x88x92fy) is the Fourier transform of c(x,y)exp[i(2xcfx80ƒx+2xcfx80ƒy)]; and
C*(xcex7+fx,xcex6+fy) is the Fourier transform of c*(x,y)exp[xe2x88x92i(2xcfx80ƒx+2xcfx80ƒy)].
Subsequently, C(xcex7xe2x88x92fx, xcex6xe2x88x92fy) is taken out by filtering, and the peak of a spectrum positioned at coordinates (fx, fy) is moved to the origin of a frequency coordinate system (also known as a Fourier spectrum coordinate system; see FIG. 8), so as to eliminate the carrier frequency. Then, inverse Fourier transform is carried out, so as to determine c(x, y), and the wrapped phase xcfx86 (x, y) is determined by the following expression (8):                               φ          ⁡                      (                          x              ,              y                        )                          =                              tan                          -              1                                ⁢                                    Im              ⁡                              [                                  c                  ⁡                                      (                                          x                      ,                      y                                        )                                                  ]                                                    Re              ⁡                              [                                  c                  ⁡                                      (                                          x                      ,                      y                                        )                                                  ]                                                                        (        8        )            
where Im[c(x, y)] is the imaginary part of c(x, y), and Re[c(x, y)] is the real part of c(x, y).
Finally, unwrapping is carried out, so as to determine the phase "PHgr"(x, y) of the object to be measured.
In the Fourier transform analysis method explained in the foregoing, the fringe image data modulated by the carrier frequency is subjected to Fourier transform as stated above.
In general, as mentioned above, the phase shift method captures the brightness of an image while imparting a phase difference between object light of an interferometer and reference light by a phase angle which is an integral fraction 2xcfx80 and analyzes thus captured brightness, thereby theoretically enabling a highly accurate phase analysis.
For securing highly accurate phase analysis, however, it is necessary that the relative relationship between a sample and a reference be displaced at a high accuracy by a predetermined phase amount (very short distance). When the phase shift method is carried out by physically moving a reference surface or the like by using a PZT (piezoelectric device), for example, it is necessary that the amount of displacement of the PZT (piezoelectric) device be controlled highly accurately. However, the displacement error of the phase shift device or the tilt error of the object surface is hard to eliminate completely, whereby controlling the amount of phase shift or tilt with a high accuracy is a difficult task in practice.
From such a viewpoint, the assignee has obtained an excellent result with a technique for detecting the above-mentioned error resulting from the phase shift device and correcting the fringe image analysis according to the detection value upon carrying out the analysis. In this technique, the fringe image data obtained by use of the phase shift method is subjected to Fourier transform, the carrier frequency and complex amplitude caused by fluctuations in wavefront occurring between an object to be observed and a reference, and the amounts of displacement and tilt of phase shift are detected according to the carrier frequency and complex amplitude, so as to correct results determined by the phase shift method, whereby influences caused by errors in the amount of tilt/displacement of the phase shift amount are eliminated.
The above-mentioned technique proposed by the assignee is a quite effective technique in that it can easily alleviate influences of errors caused by the phase shift device without using an expensive phase shift device. However, since calculations for correcting the amount of error from a predetermined step amount (e.g., 90 degrees in the case of 4-bucket (step) method), the time required for fringe analysis becomes longer. Also, even in the case where 3-bucket method is theoretically sufficient when no shift error exists in the phase shift device, it is necessary to use 4-bucket method or 5-bucket method from the viewpoint of accuracy, which may also hinder the time required for fringe analysis from becoming shorter.
In view of the foregoing circumstances, it is an object of the present invention to provide a phase shift fringe image analysis method which can eliminate influences caused by errors in the amount of displacement of phase shift and/or relative tilt amounts between the object and the reference without making the apparatus configuration complicated and expensive when analyzing fringe image data obtained by using a phase shift method, whereby the fringe analysis can be carried out rapidly and favorably; and an apparatus using the same.
The present invention provides a phase shift fringe image analysis method comprising the steps of shifting an object to be observed and a reference relative to each other by using a phase shift device, obtaining fringe image data at a plurality of phase shift positions, and determining a phase of the object by analyzing thus obtained plurality of fringe image data items;
wherein the plurality of phase shift positions are at least three phase positions having a given phase gap therebetween; and
wherein positional data of the above-mentioned at least three phase positions are specified, and the whole or part of the fringe image data on which carrier fringes at these phase positions are superposed is subjected to a predetermined arithmetic operation so as to carry out a phase analysis and determine the phase of the object.
The positional data of the above-mentioned at least three phase positions may be determined by a Fourier transform fringe analysis method.
The predetermined arithmetic operation may be carried out in view of data concerning relative tilt between the object and the reference at the above-mentioned at least three phase positions.
The data concerning relative tilt between the object and the reference may be determined from a difference in frequency of the carrier fringes.
The data concerning relative tilt between the object and the reference may be determined from a difference in phase of the object.
The number of phase shift positions for determining the fringe image data may be 3, and the phase of the object may be represented by the following conditional expression (1):                                           φ            ⁡                          (                              x                ,                y                            )                                =                                    arctan              ⁢              cos              ⁢                              xe2x80x83                            ⁢                              δ                3                                      -                                          (                                  1                  +                  p                                )                            ⁢              cos              ⁢                              xe2x80x83                            ⁢                              δ                2                                      +                                          p                ⁢                                  xe2x80x83                                ⁢                cos                ⁢                                  xe2x80x83                                ⁢                                  δ                  1                                                                              sin                  ⁢                                      xe2x80x83                                    ⁢                                      δ                    3                                                  -                                                      (                                          1                      +                      p                                        )                                    ⁢                  sin                  ⁢                                      xe2x80x83                                    ⁢                                      δ                    2                                                  +                                  p                  ⁢                                      xe2x80x83                                    ⁢                  sin                  ⁢                                      xe2x80x83                                    ⁢                                      δ                    1                                                                                      ⁢                  
                ⁢                  xe2x80x83                ⁢        where        ⁢                  
                ⁢                                                                              p                  =                                                                                                              i                          3                                                -                                                  i                          2                                                                                                                      i                          2                                                -                                                  i                          1                                                                                      =                                        ⁢                                                                                            cos                          ⁡                                                      [                                                                                          φ                                ⁡                                                                  (                                                                      x                                    ,                                    y                                                                    )                                                                                            +                                                              δ                                3                                                                                      ]                                                                          -                                                  cos                          ⁡                                                      [                                                                                          φ                                ⁡                                                                  (                                                                      x                                    ,                                    y                                                                    )                                                                                            +                                                              δ                                2                                                                                      ]                                                                                                                                                cos                          ⁡                                                      [                                                                                          φ                                ⁡                                                                  (                                                                      x                                    ,                                    y                                                                    )                                                                                            +                                                              δ                                2                                                                                      ]                                                                          -                                                  cos                          ⁡                                                      [                                                                                          φ                                ⁡                                                                  (                                                                      x                                    ,                                    y                                                                    )                                                                                            +                                                              δ                                1                                                                                      ]                                                                                                                                              ,                                  xe2x80x83                                ⁢                and                                                                                                                              i                    m                                    ⁡                                      (                                          x                      ,                      y                      ,                                              ξ                        m                                                              )                                                  =                                ⁢                                                      a                    ⁡                                          (                                              x                        ,                        y                                            )                                                        +                                                            b                      ⁡                                              (                                                  x                          ,                          y                                                )                                                              ⁢                                          cos                      ⁡                                              [                                                                              2                            ⁢                            π                            ⁢                                                          xe2x80x83                                                        ⁢                                                          f                              xm                                                        ⁢                            x                                                    +                                                      2                            ⁢                            π                            ⁢                                                          xe2x80x83                                                        ⁢                                                          f                              ym                                                        ⁢                            y                                                    +                                                      φ                            ⁡                                                          (                                                              x                                ,                                y                                                            )                                                                                +                                                      ξ                            m                                                                          ]                                                                                                                                                                    =                                ⁢                                                      a                    ⁡                                          (                                              x                        ,                        y                                            )                                                        +                                                            b                      ⁡                                              (                                                  x                          ,                          y                                                )                                                              ⁢                                          cos                      ⁡                                              [                                                                              φ                            ⁡                                                          (                                                              x                                ,                                y                                                            )                                                                                +                                                      δ                            m                                                                          ]                                                                                                                                                    (        3        )            
where
a(x, y) is the background of interference fringes;
b(x, y) is the visibility of fringes;
xcfx86 (x, y) is the phase of the object; and
xcex4m is the phase shift amount of the phase shift device expressed by:
xcex4m=2xcfx80ƒxmx+2xcfx80ƒymy+"xgr"m 
where
"xgr"m is the phase of the phase shift device (not including the part involved with the tilt of the phase shift device); and
fxm and fym are the carrier frequencies (including the part of the error in inclination of the phase shift device) after the m-th phase shift expressed by:             ξ      m        =          2      ⁢      π      ⁢                        z          m                λ              ,      xe2x80x83    ⁢            f      xm        =                            2          ·          tan                ⁢                  xe2x80x83                ⁢                  θ          xm                    λ        ,      xe2x80x83    ⁢            f      ym        =                            2          ·          tan                ⁢                  xe2x80x83                ⁢                  θ          ym                    λ      
where
xcex is the wavelength of light;
xcex8xm and xcex8ym are respective inclinations of the object surface upon the m-th phase shift in x and y directions; and
zm is the amount of displacement of the phase shift device at the m-th shift position (not including the part involved with the tilt of the phase shift device).
The phase shift fringe analysis method in accordance with the present invention may comprise the steps of determining a complex amplitude of a fringe image by the Fourier transform fringe analysis method, and obtaining the above-mentioned at least three phase positions according to thus determined complex amplitude.
The phase shift fringe analysis method in accordance with the present invention may comprise the steps of selecting a plurality of sets of at least three local fringe image data items corresponding to each other from fringe image data at the above-mentioned at least three phase shift positions, obtaining positional data of the above-mentioned at least three phase positions concerning each set according to the fringe image data of respective set, and averaging positional data of phase positions by a number corresponding to the number of the sets, so as to determine final positional data of the above-mentioned at least three phase positions.
The fringe image may be an interference fringe image.
The present invention provides a phase shift fringe analysis apparatus for shifting an object to be observed and a reference relative to each other by using a phase shift device, obtaining fringe image data at a plurality of phase shift positions, and analyzing thus obtained plurality of fringe image data items so as to determine a phase of the object;
wherein the plurality of phase shift positions are at least three phase positions having a given phase gap therebetween; and
wherein the apparatus comprises data acquiring means for obtaining positional data of the at least three phase positions, and phase analysis means for carrying out a phase analysis by subjecting the whole or part of the fringe image data on which carrier fringes at these phase positions are superposed to a predetermined arithmetic operation.